Q46. - Algebra 2

Question

For Exercises 45-47, use the following information.

Triangle $\mathbf{ABC}$with vertices $\mathbf{A}\left(\mathbf{0}\mathbf{,}\mathbf{2}\right)$, $\mathbf{B}\left(\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{1}\right)$ and $\mathbf{C}\left(\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{4}\right)$ is translated 1 unit right and 3 units up.

46. Find the coordinates of $\mathbf{\Delta A}\mathbf{\text{'}}\mathbf{B}\mathbf{\text{'}}\mathbf{C}\mathbf{\text{'}}$.

Step-by-Step Solution

Answer

The coordinates of $\Delta A\text{'}B\text{'}C\text{'}$ are: $\mathbf{A}\mathbf{\text{'}}\left(\mathbf{1}\mathbf{,}\mathbf{5}\right)$, $\mathbf{B}\mathbf{\text{'}}\left(\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{2}\right)$ and $\mathbf{C}\mathbf{\text{'}}\left(\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{1}\right)$.

Step 1 ­- Description of step.

A translation of a figure is said to occur when a figure is moved from one location to other without having change in its size, shape or orientation.

Step 2 ­- Write the vertex matrix for the triangle ABC.

The vertex matrix for the triangle $ABC$ is given by:$\left[\begin{array}{ccc}0& -3& -2\\ 2& -1& -4\end{array}\right]$

Step 3 ­- Description of step.

As the given triangle is translated 1 unit right and 3 units up therefore to translate the given figure, add $1$ to each $x$-coordinate and add 3 to each $y$-coordinate.

Therefore the translation matrix has each column as $\left[\begin{array}{c}1\\ 3\end{array}\right]$.

Therefore, the translation matrix is given by:$\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& 3\end{array}\right]$

Step 4 ­- Find the coordinates of ΔA ' B ' C ' .

The vertex matrix of the triangle $\Delta A\text{'}B\text{'}C\text{'}$ can be find out by adding the vertex matrix for the triangle $ABC$with the translation matrix.

Therefore, the vertex matrix of the triangle $\Delta A\text{'}B\text{'}C\text{'}$ is given by:

$\begin{array}{c}\text{vertex matrix for ΔA'B'C}\text{'}=\left[\begin{array}{ccc}0& -3& -2\\ 2& -1& -4\end{array}\right]+\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& 3\end{array}\right]\\ =\left[\begin{array}{ccc}0+1& -3+1& -2+1\\ 2+3& -1+3& -4+3\end{array}\right]\\ =\left[\begin{array}{ccc}1& -2& -1\\ 5& 2& -1\end{array}\right]\end{array}$

Therefore, the coordinates of the triangles $\Delta A\text{'}B\text{'}C\text{'}$ are: $A\text{'}\left(1,5\right)$, $B\text{'}\left(-2,2\right)$ and $C\text{'}\left(-1,-1\right)$.